206. Choosing Models

Learning Intentions

• Choose a linear or quadratic model for a given situation.
• Justify model choice Use patterns in data.
• Explain limitations of a chosen model.

Pre-requisite Summary

• Know that a linear model has constant first differences and can often be written in the form .
• Know that a linear graph is a straight line.
• Know that a quadratic model has changing first differences and often constant second differences.
• Know that a quadratic graph is a parabola and can often model area, height or curved motion.
• Know that a model is a simplified mathematical description of a situation.
• Know that models may only be accurate over a certain range of values or under certain assumptions.

Worked Examples

Worked Example 1

Choose whether a linear or quadratic model is more appropriate.

A parking garage charges $ for entry and $ for each hour parked.

a) State whether the model should be linear or quadratic.

b) Explain what feature of the situation supports your choice.

c) Write a possible rule for the total cost after hours.

Worked Example 2

Choose whether a linear or quadratic model is more appropriate.

A square has side length metres and area square metres.

a) State whether the model should be linear or quadratic.

b) Explain what feature of the situation supports your choice.

c) Write a possible rule for the area.

Worked Example 3

Use the table to choose a linear or quadratic model.

a) Find the first differences.

b) Choose a linear or quadratic model.

c) Justify your choice using the pattern in the data.

Worked Example 4

Use the table to choose a linear or quadratic model.

a) Find the first differences.

b) Find the second differences.

c) Choose a linear or quadratic model and justify your choice.

Worked Example 5

A ball is thrown into the air. Its height is modelled by , where is height in metres and is time in seconds.

a) Explain why a quadratic model is appropriate.

b) Interpret one key feature of the model.

c) State one limitation of the model.

Worked Example 6

A student models the cost of a phone plan using , where is total cost in dollars and is the number of months.

a) Explain why a linear model is appropriate.

b) Interpret the gradient and intercept.

c) State one limitation of the model.

Worked Example 7

A rectangle has length metres and width metres.

a) Write a rule for the area .

b) Choose whether the model is linear or quadratic.

c) Explain one limitation of using this model in a real garden design.

Problems

Problem 1

Choose whether a linear or quadratic model is more appropriate.

A streaming service charges $ to join and $ each month.

a) State whether the model should be linear or quadratic.

b) Explain what feature of the situation supports your choice.

c) Write a possible rule for the total cost after months.

Problem 2

Choose whether a linear or quadratic model is more appropriate.

A square has side length centimetres and area square centimetres.

a) State whether the model should be linear or quadratic.

b) Explain what feature of the situation supports your choice.

c) Write a possible rule for the area.

Problem 3

Use the table to choose a linear or quadratic model.

a) Find the first differences.

b) Choose a linear or quadratic model.

c) Justify your choice using the pattern in the data.

Problem 4

Use the table to choose a linear or quadratic model.

a) Find the first differences.

b) Find the second differences.

c) Choose a linear or quadratic model and justify your choice.

Problem 5

A ball is thrown into the air. Its height is modelled by , where is height in metres and is time in seconds.

a) Explain why a quadratic model is appropriate.

b) Interpret one key feature of the model.

c) State one limitation of the model.

Problem 6

A student models the cost of a gym membership using , where is total cost in dollars and is the number of weeks.

a) Explain why a linear model is appropriate.

b) Interpret the gradient and intercept.

c) State one limitation of the model.

Problem 7

A rectangle has length metres and width metres.

a) Write a rule for the area .

b) Choose whether the model is linear or quadratic.

c) Explain one limitation of using this model in a real garden design.

Exercises

Understanding and Fluency

Exercise 1

Choose whether a linear or quadratic model is more appropriate.

a) A taxi charges $ plus $ per kilometre.

b) The area of a square changes as its side length changes.

c) A ball is thrown upward and falls back down.

d) A worker earns $ per hour.

Exercise 2

Choose whether a linear or quadratic model is more appropriate.

a) A candle loses cm of height each hour.

b) A rectangle has width and length .

c) A phone plan costs $ plus $ per month.

d) The height of water from a fountain follows a curved path.

Exercise 3

For each table, decide whether the pattern is linear or quadratic.

a)

b)

Exercise 4

Find the first differences and use them to choose a model.

a)

b)

Exercise 5

For each table, find the first differences and second differences.

a)

b)

Exercise 6

Match each context to the most appropriate model type.

a) Constant weekly savings

b) Area of a square

c) Height of a thrown ball

d) Cost with a fixed starting fee and a constant charge

Model types:

• linear
• quadratic

Exercise 7

For each rule, decide whether it represents a linear or quadratic model.

a)

b)

c)

d)

Exercise 8

For each model, identify one key feature.

a) , where is cost after months.

b) , where is the area of a square with side length .

c) , where is height after seconds.

Exercise 9

State one possible limitation of each model.

a) A taxi fare model .

b) A ball height model .

c) A square area model .

Exercise 10

Complete each sentence.

a) A linear model is appropriate when the first differences are __________.

b) A quadratic model is often appropriate when the second differences are WHAT.

c) A model may be limited because it only works for a certain WHAT of values.

Reasoning

Exercise 11

Explain why constant first differences suggest a linear model.

Exercise 12

Explain why constant second differences suggest a quadratic model.

Exercise 13

A student says this table is linear because the -values increase.

Explain the mistake and choose a better model.

Exercise 14

A student says a phone plan should be modelled by a quadratic because the total cost gets larger each month.

Explain why a linear model may be more appropriate.

Exercise 15

A rectangular area is modelled by .

a) Explain why the model is quadratic.

b) Explain why cannot be negative in this context.

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) A linear model always has constant first differences.

b) A quadratic model always has constant first differences.

c) A model can be useful even if it is not perfect.

Problem-solving

Exercise 17

A student records the total cost of hiring a bike.

Hours
Cost	$	$	$	$	$

a) Find the first differences.

b) Choose a linear or quadratic model.

c) Write a rule for the cost after hours.

d) State one limitation of the model.

Exercise 18

A student records the area of a square.

Side length
Area

a) Find the first differences.

b) Find the second differences.

c) Choose a linear or quadratic model.

d) State one limitation of the model.

Exercise 19

A ball’s height is recorded after it is thrown upward.

Time
Height

a) Explain why the data is not linear.

b) Choose a more appropriate model type.

c) Describe one key feature the model should have.

d) State one limitation of using the model.

Exercise 20

Create your own modelling example.

Your response must include:

• a real-world situation
• a table of at least five values
• a decision about whether a linear or quadratic model is more appropriate
• a justification using first differences or second differences
• one limitation of the chosen model

Potential Misunderstandings

• Students may choose a linear model whenever the values increase, even if the increase is not constant.

• Students may choose a quadratic model whenever a graph curves, without checking whether the context or data supports it.

• Students may ignore the meaning of the variables when selecting a model.

• Students may calculate first differences incorrectly by subtracting non-adjacent values.

• Students may identify changing first differences but forget to check second differences.

• Students may assume that a table with only a few values proves a model is correct for all values.

• Students may think a model must be perfectly accurate to be useful.

• Students may forget that context can restrict input values, such as time, length or money not being negative.

• Students may use a model outside its sensible domain, such as predicting a negative height or cost.

Next: 207. Building Models from Data